Optimal. Leaf size=45 \[ \frac{a x}{c \sqrt{c+\frac{d}{x^2}}}-\frac{b c-2 a d}{c^2 x \sqrt{c+\frac{d}{x^2}}} \]
[Out]
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Rubi [A] time = 0.103355, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{a x}{c \sqrt{c+\frac{d}{x^2}}}-\frac{b c-2 a d}{c^2 x \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)/(c + d/x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 10.7126, size = 37, normalized size = 0.82 \[ \frac{a x}{c \sqrt{c + \frac{d}{x^{2}}}} + \frac{2 a d - b c}{c^{2} x \sqrt{c + \frac{d}{x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)/(c+d/x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0339211, size = 33, normalized size = 0.73 \[ \frac{a c x^2+2 a d-b c}{c^2 x \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)/(c + d/x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.008, size = 43, normalized size = 1. \[{\frac{ \left ( a{x}^{2}c+2\,ad-bc \right ) \left ( c{x}^{2}+d \right ) }{{x}^{3}{c}^{2}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)/(c+d/x^2)^(3/2),x)
[Out]
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Maxima [A] time = 1.423, size = 72, normalized size = 1.6 \[ a{\left (\frac{\sqrt{c + \frac{d}{x^{2}}} x}{c^{2}} + \frac{d}{\sqrt{c + \frac{d}{x^{2}}} c^{2} x}\right )} - \frac{b}{\sqrt{c + \frac{d}{x^{2}}} c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/(c + d/x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232242, size = 63, normalized size = 1.4 \[ \frac{{\left (a c x^{3} -{\left (b c - 2 \, a d\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c^{3} x^{2} + c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/(c + d/x^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.101, size = 65, normalized size = 1.44 \[ a \left (\frac{x^{2}}{c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{2 \sqrt{d}}{c^{2} \sqrt{\frac{c x^{2}}{d} + 1}}\right ) - \frac{b}{c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)/(c+d/x**2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/(c + d/x^2)^(3/2),x, algorithm="giac")
[Out]